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What is this series? What is its analytical solution?

$$x_{n+1}=\frac{A+Bx_{n}}{C+Dx_{n}}$$

This can also be written as:

$$x_{n+1}=\frac{B}{D}+\frac{A-\frac{BC}{D}}{C+Dx_{n}}$$

Parsa Rahimi
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  • Are you looking for the limit $n\to \infty$? – Andrei Jul 12 '22 at 20:37
  • I'd say it will very much depend on the coefficients, you can have no fixed points or 1 or 2, repulsive or attractive, some converging in staircase ($x_n$ monotonous) other in spiral ($x_{2n}$ and $x_{2n+1}$ monotonous). Here are some specific examples I treated, notice the technique is similar. https://math.stackexchange.com/a/2624102/399263, https://math.stackexchange.com/a/2522776/399263, https://math.stackexchange.com/a/3823634/399263. – zwim Jul 12 '22 at 20:43
  • No actually I was hoping for an analytical solution – Parsa Rahimi Jul 12 '22 at 20:43
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    Hint, this is a Moebius transformation and so can be represented by a matrix. Then you can generate iterates by considering powers of the matrix, we can in turn be found by diagonalizing. – Jair Taylor Jul 12 '22 at 20:47
  • @Jair yep, my second link has a showcase of that method by Robert Israel. – zwim Jul 12 '22 at 20:51
  • @zwim I'd like to get your name and give credit for helping me out, I am writing a paper. – Parsa Rahimi Jul 12 '22 at 22:12
  • @JairTaylor I'd like to get your name and give credit for helping me out, I am writing a paper. – Parsa Rahimi Jul 12 '22 at 22:12
  • @ParsaRahimi Thank you but it's really not necessary, this is a well-known and standard technique. In any case this is my real name. – Jair Taylor Jul 12 '22 at 22:13

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In my answer to this question, I detailed the steps for solving a first-order rational difference equation such as $${ x_{n+1} = \frac{a+m\,x_n }{b+x_n } }$$

So, for your case, let $a=\frac AD$, $m=\frac BD$ and $b=\frac CD$

Claude Leibovici
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